You can fly to Cyprus from most UK airports. The flying time is between 4 and 5 hours and you should take into account that Cyprus is 2 hours ahead of UK time. There are two international airports in Cyprus.
The closest to the holiday apartment is Paphos which is a 20min drive and costs about 15CYP in a taxi.
The other airport is Larnaca which is a 90min drive from the holiday apartment and costs about 50CYP.
If you want to book your flights on-line there are some links below to help.